Near-optimality of $\Sigma\Delta$ quantization for $L^2$-approximation with polynomials in Bernstein form

Sinan Gunturk, Weilin Li

Abstract: In this paper, we provide lower bounds on the $L^2$-error of approximation of arbitrary functions $f: [0,1] \to \mathbb{R}$ by polynomials of degree at most $n$, with the constraint that the coefficients of these polynomials in the Bernstein basis of order $n$ are bounded by $n^\alpha$ for some $\alpha \geq 0$. For Lipschitz functions, this lower bound matches, up to a factor of $\sqrt{\log n}$, a previously obtained upper bound for the error of approximation by one-bit polynomials in Bernstein form via $\Sigma\Delta$ quantization where the functions are bounded by $1$ and the coefficients of the approximating polynomials are constrained to be in $\{\pm 1\}$.

Submission Type: Full Paper